Central London is very expensive. Try some areas which are at some distance but not too expensive. Having an excellent transport system, you won't find any trouble to move around. Say you can go for Walthamstow Central or Stratford, nice areas and hardly 20-25 min commute from central london. You will find a big difference.
I see you are interested to visit Trafalgar Square and Piers near Thames, but don't forget Natural History Museum, it will be a real treat for you and your child.
For scotland, you can opt for Guesthouse. I stayed at Gilmore Guest house, it was 40GBP per day but was good for family. As someone else has mentioned, in the UK the early you book the more you save, whether it be Train tickets or it be guest house/hotels.